Question

Consider the function f(x)=arctan [(x+6)/(x+5)]

Express the domain of the function in interval notation: Find the y-intercept: y= . Find all the x-intercepts (enter your answer as a comma-separated list): x= . Does f have any symmetries? f is even; f is odd; f is periodic; None of the above. Find all the asymptotes of f (enter your answers as comma-separated list; if the list is empty, enter DNE): Vertical asymptotes: ; Horizontal asymptotes: ; Slant asymptotes: . Determine the derivative of f. f'(x)= On which intervals is f increasing/decreasing? (Use the union symbol and not a comma to separate different intervals; if the function is nowhere increasing or nowhere decreasing, use DNE as appropriate). f is increasing on . f is decreasing on . List all the local maxima and minima of f. Enter each maximum or minimum as the coordinates of the point on the graph. For example, if f has a maximum at x=3 and f(3)=9, enter (3,9) in the box for maxima. If there are multiple maxima or minima, enter them as a comma-separated list of points, e.g. (3,9),(0,0),(4,7) . If there are none, enter DNE. Local maxima: . Local minima: . Determine the second derivative of f. f''(x)= On which intervals does f have concavity upwards/downwards? (Use the union symbol and not a comma to separate different intervals; if the function does not have concavity upwards or downwards on any interval, use DNE as appropriate). f is concave upwards on . f is concave downwards on . List all the inflection points of f. Enter each inflection point as the coordinates of the point on the graph. For example, if f has an inflection point at x=7 and f(7)=−2, enter (7,−2) in the box. If there are multiple inflection points, enter them as a comma-separated list, e.g. (7,−2),(0,0),(4,7) . If there are none, enter DNE.

Does the function have any of the following features? Select all that apply.

Removable discontinuities (i.e. points where the limit exists, but it is different than the value of the function)

Corners (i.e. points where the left and right derivatives are defined but are different)

Jump discontinuities (i.e. points where the left and right limits exist but are different)

Points with a vertical tangent line

Upload a sketch of the graph of f. You can use a piece of paper and a scanner or a camera, or you can use a tablet, but the sketch must be drawn by hand. You should include all relevant information that has not been requested here, for example the limits at the edges of the domain and the slopes of tangent lines at interesting points (e.g. inflection points). Make sure that the picture is clear, legible, and correctly oriented.

Answer 1

Point of inflection occur if graph changes concavity, but this function is monotonic decreasing function. Thus no point of inflection or critical point.

(discriminant of quadratic equation in denominator of f'(x) is less than zero, thus always negative. This makes f'(x) < 0)

This function is concave down from (- infinity to ) and concave up from (, infinity)

Maximum value of it is when x is , and

minimum value of it is when x is

Thus, It has no point/removable discontinuity, it has jump discontinuity at x = -5.

Corners are : ( ,  ) and ( ,  )

Graph: